Admissibility and polynomial dichotomies for evolution families

Davor Dragicevic

arxiv: 1907.02515 · v1 · pith:MYDOQHRXnew · submitted 2019-07-02 · 🧮 math.DS

Admissibility and polynomial dichotomies for evolution families

Davor Dragicevic This is my paper

Pith reviewed 2026-05-25 10:32 UTC · model grok-4.3

classification 🧮 math.DS

keywords evolution familiespolynomial dichotomiesadmissibilityLyapunov normsnonuniform polynomial dichotomiesrobustnesslinear perturbations

0 comments

The pith

Polynomial dichotomies for evolution families are equivalent to admissibility of bounded perturbations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that an evolution family admits a polynomial dichotomy with respect to a family of norms precisely when it satisfies the admissibility property. Admissibility requires that every bounded perturbation of the family has a unique bounded solution. Using Lyapunov norms as the family recovers the notion of a strong nonuniform polynomial dichotomy. The characterization then shows that this dichotomy property remains intact under small linear perturbations to the evolution family.

Core claim

For an arbitrary evolution family, the notion of a polynomial dichotomy with respect to a family of norms is characterized in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, the notion of a (strong) nonuniform polynomial dichotomy is recovered. The characterization is used to establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.

What carries the argument

The admissibility property: for every bounded perturbation there is a unique bounded solution.

If this is right

The equivalence recovers nonuniform polynomial dichotomies when Lyapunov norms are used.
Strong nonuniform polynomial dichotomies persist under small linear perturbations.
Verification of polynomial dichotomies can proceed by checking the existence of unique bounded solutions rather than constructing splitting projections directly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This suggests that similar admissibility characterizations could apply to other dichotomy notions like exponential dichotomies.
Such results may aid in analyzing stability for nonautonomous differential equations in applications.
The robustness result implies that small modeling errors do not destroy the dichotomy property.

Load-bearing premise

A suitable family of norms exists with respect to which both the dichotomy and admissibility are defined.

What would settle it

Construct an evolution family and a family of norms where there is a unique bounded solution for every bounded perturbation but the polynomial dichotomy fails to hold.

read the original abstract

For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a (strong) nonuniform polynomial dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of a strong nonuniform polynomial dichotomy under sufficiently small linear perturbations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a characterization of polynomial dichotomies via admissibility for evolution families w.r.t. a given family of norms, recovers the strong nonuniform case with Lyapunov norms, and proves robustness under small linear perturbations.

read the letter

The core contribution is a direct equivalence between polynomial dichotomies (relative to a prescribed family of norms) and the admissibility property for arbitrary evolution families, plus the recovery of the nonuniform version and a robustness corollary. This is a standard structural move in dichotomy theory, but the extension to the polynomial case with the nonuniform recovery via Lyapunov norms looks like the actual new statements. The robustness application is presented as nontrivial, which fits the pattern of using admissibility to get perturbation results without extra work. The setup treats the family of norms as given rather than derived, which is explicit in the abstract and avoids hidden assumptions. No circularity or fitting issues appear. The main limitation is that everything is relative to the chosen norms, so readers interested in constructing dichotomies from the evolution family alone will still need to find or build those norms separately. This is common in the subfield and not a flaw in the stated claims. The paper is aimed at specialists working on nonautonomous systems, stability, and dichotomy theory. It is the kind of precise equivalence result that deserves referee time if the proofs are complete and the assumptions are handled cleanly. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper claims that for an arbitrary evolution family, the notion of a polynomial dichotomy with respect to a given family of norms is equivalent to an admissibility property (unique bounded solution for every bounded perturbation). Using a family of Lyapunov norms recovers the strong nonuniform polynomial dichotomy, and the characterization is applied to prove robustness of strong nonuniform polynomial dichotomies under sufficiently small linear perturbations.

Significance. If the equivalence holds, the result supplies a standard but useful tool for establishing polynomial dichotomies via admissibility, which is often more tractable than direct estimates. The recovery of the nonuniform case via Lyapunov norms and the robustness corollary constitute nontrivial extensions within dichotomy theory for nonautonomous systems. The approach treats the family of norms as part of the given data rather than deriving it from the evolution family alone.

minor comments (2)

[Abstract] Abstract: the statement does not list the standing assumptions on the evolution family or on the family of norms; adding one sentence would clarify the setup without lengthening the abstract.
The notation for the family of norms and the precise definition of polynomial dichotomy should be introduced with an explicit reference to the underlying Banach space and time interval at the first occurrence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment. The referee's summary correctly identifies the main results: the admissibility characterization for polynomial dichotomies with respect to a given family of norms, the recovery of strong nonuniform polynomial dichotomies via Lyapunov norms, and the robustness corollary. We are pleased with the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes a direct equivalence between polynomial dichotomy (w.r.t. a supplied family of norms) and the admissibility property for arbitrary evolution families, then specializes to Lyapunov norms to recover the nonuniform case and derives a robustness corollary. No quoted step reduces a claimed prediction or uniqueness result to a fitted input, self-citation chain, or definitional renaming; the norms are explicitly part of the given setup rather than derived from the evolution family alone. The central result is therefore a self-contained structural characterization with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no concrete free parameters, axioms, or invented entities can be extracted. The central claims rest on the existence of suitable families of norms and on standard properties of evolution families that are not detailed here.

pith-pipeline@v0.9.0 · 5593 in / 1057 out tokens · 48709 ms · 2026-05-25T10:32:04.823844+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 7: robustness under ||B(t)|| ≤ c/t^{1+ε}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

Barreira, D

L. Barreira, D. Dragiˇ cevi´ c and C. Valls,Strong and weak (Lp, Lq)-admissibility, Bull. Sci. Math. 138 (2014), 721–741

work page 2014
[2]

Barreira, D

L. Barreira, D. Dragiˇ cevi´ c and C. Valls,Admissibility on the half line for evo- lution families , J. Anal. Math. 132 (2017), 157–176

work page 2017
[3]

Barreira, D

L. Barreira, D. Dragiˇ cevi´ c and C. Valls, Admissibility and hyperbolicity , Springer Briefs in Mathematics (2018), Springer

work page 2018
[4]

Barreira and C

L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity , Discrete Contin. Dynam. Syst. 22 (2008), 509–528

work page 2008
[5]

Barreira and C

L. Barreira and C. Valls, Polynomial growth rates , Nonlinear Anal. 71 (2009), 5208–5219

work page 2009
[6]

Barreira and C

L. Barreira and C. Valls, Robustness of noninvertible dichotomies , J. Math. Soc. Japan 67 (2015), 293–317

work page 2015
[7]

Bento and C

A. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies , J. Funct. Anal. 257 (2009), 122–148

work page 2009
[8]

Bento and C

A. Bento and C. Silva, Stable manifolds for nonautonomous equations with nonuniform polynomial dichotomies , Q. J. Math 63 (2012), 275–308

work page 2012
[9]

Coppel, Dichotomies in Stability Theory, Lect

W. Coppel, Dichotomies in Stability Theory, Lect. Notes in Math. 62 9, Springer, 1978

work page 1978
[10]

Dalec ′ki ˘ ı and M

Ju. Dalec ′ki ˘ ı and M. Kre ˘ ın,Stability of Solutions of Diﬀerential Equations in Banach Space , Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974

work page 1974
[11]

Dragiˇ cevi´ c, A spectral characterization of nonuniform polynomial di- chotomies, Math

D. Dragiˇ cevi´ c, A spectral characterization of nonuniform polynomial di- chotomies, Math. Nachr., to appear

work page
[12]

P. V. Hai, On the polynomial stability of evolution families , Appl. Anal. 95 (2016), 1239–1255

work page 2016
[13]

Henry, Geometric Theory of Semilinear Parabolic Equations , Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981

D. Henry, Geometric Theory of Semilinear Parabolic Equations , Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981

work page 1981
[14]

Huy, Exponential dichotomy of evolution equations and admissib ility of func- tion spaces on a half-line , J

N. Huy, Exponential dichotomy of evolution equations and admissib ility of func- tion spaces on a half-line , J. Funct. Anal. 235 (2006), 330–354. ADMISSIBILITY AND POLYNOMIAL DICHOTOMIES 19

work page 2006
[15]

Latushkin, T

Y. Latushkin, T. Randolph and R. Schnaubelt, Exponential dichotomy and mild solution of nonautonomous equations in Banach spaces , J. Dynam. Diﬀerential Equations 10 (1998), 489–510

work page 1998
[16]

Li, Die Stabilit¨ atsfrage bei Diﬀerenzengleichungen, Acta Math

T. Li, Die Stabilit¨ atsfrage bei Diﬀerenzengleichungen, Acta Math. 63 (1934), 99–141

work page 1934
[17]

Lupa and L

N. Lupa and L. Popescu, Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line , Semigroup Forum 98 (2019), 184– 208

work page 2019
[18]

Massera and J

J. Massera and J. Sch¨ aﬀer,Linear diﬀerential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517–573

work page 1958
[19]

Massera and J

J. Massera and J. Sch¨ aﬀer, Linear Diﬀerential Equations and Function Spaces , Pure and Applied Mathematics 21, Academic Press, New York-Londo n, 1966

work page 1966
[20]

Megan, A

M. Megan, A. L. Sasu and B. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces , Integral Equations Operator Theory 44 (2002), 71–78

work page 2002
[21]

J. S. Muldowney, Dichotomies and asymptotic behaviour for linear diﬀerenti al systems, Trans. Amer. Math. Soc. 283 (1984), 465–484

work page 1984
[22]

Naulin and M

R. Naulin and M. Pinto, Roughness of (h, k)-dichotomies, J. Diﬀerential Equa- tions 118 (1995), 20–35

work page 1995
[23]

Naulin and M

R. Naulin and M. Pinto, Stability of Discrete Dichotomies for Linear Diﬀerence Systems, J. Diﬀerence Equ. Appl. 3 (1997), 101–123

work page 1997
[24]

Perron, Die Stabilit¨ atsfrage bei Diﬀerentialgleichungen, Math

O. Perron, Die Stabilit¨ atsfrage bei Diﬀerentialgleichungen, Math. Z. 32 (1930), 703–728

work page 1930
[25]

Preda and M

P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc. 27 (1983), 31–52

work page 1983
[26]

Preda, A

P. Preda, A. Pogan and C. Preda, ( Lp, Lq)-admissibility and exponential di- chotomy of evolutionary processes on the half-line , Integral Equations Operator Theory 49 (2004), 405–418

work page 2004
[27]

Preda, A

P. Preda, A. Pogan and C. Preda, Sch¨ aﬀer spaces and exponential dichotomy for evolutionary processes , J. Diﬀerential Equations 230 (2006), 378–391

work page 2006
[28]

A. L. Sasu, M. Babutia and B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line , Bull. Sci. Math. 137 (2013), 466–484

work page 2013
[29]

A. L. Sasu and B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces , Integral Equations Operator Theory 54 (2006), 113–130

work page 2006
[30]

A. L. Sasu and B. Sasu, Exponential trichotomy and p-admissibility for evolu- tion families on the real line , Math. Z. 253 (2006), 515–536

work page 2006
[31]

A. L. Sasu and B. Sasu, Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications , Integral Equations Operator Theory 66 (2010), 113–140

work page 2010
[32]

Van Minh, F

N. Van Minh, F. R¨ abiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equ ations on the half- line, Integral Equations Operator Theory 32 (1998), 332–353

work page 1998
[33]

Zhou and W

L. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for diﬀerence equations , J. Funct. Anal. 271 (2016), 1087–1129

work page 2016
[34]

L. Zhou, K. Lu and W. Zhang, Equivalences between nonuniform exponential dichotomy and admissibility , J. Diﬀerential Equations 262 (2017), 682–747. Department of Mathematics, University of Rijeka, Croatia E-mail address : ddragicevic@math.uniri.hr

work page 2017

[1] [1]

Barreira, D

L. Barreira, D. Dragiˇ cevi´ c and C. Valls,Strong and weak (Lp, Lq)-admissibility, Bull. Sci. Math. 138 (2014), 721–741

work page 2014

[2] [2]

Barreira, D

L. Barreira, D. Dragiˇ cevi´ c and C. Valls,Admissibility on the half line for evo- lution families , J. Anal. Math. 132 (2017), 157–176

work page 2017

[3] [3]

Barreira, D

L. Barreira, D. Dragiˇ cevi´ c and C. Valls, Admissibility and hyperbolicity , Springer Briefs in Mathematics (2018), Springer

work page 2018

[4] [4]

Barreira and C

L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity , Discrete Contin. Dynam. Syst. 22 (2008), 509–528

work page 2008

[5] [5]

Barreira and C

L. Barreira and C. Valls, Polynomial growth rates , Nonlinear Anal. 71 (2009), 5208–5219

work page 2009

[6] [6]

Barreira and C

L. Barreira and C. Valls, Robustness of noninvertible dichotomies , J. Math. Soc. Japan 67 (2015), 293–317

work page 2015

[7] [7]

Bento and C

A. Bento and C. Silva, Stable manifolds for nonuniform polynomial dichotomies , J. Funct. Anal. 257 (2009), 122–148

work page 2009

[8] [8]

Bento and C

A. Bento and C. Silva, Stable manifolds for nonautonomous equations with nonuniform polynomial dichotomies , Q. J. Math 63 (2012), 275–308

work page 2012

[9] [9]

Coppel, Dichotomies in Stability Theory, Lect

W. Coppel, Dichotomies in Stability Theory, Lect. Notes in Math. 62 9, Springer, 1978

work page 1978

[10] [10]

Dalec ′ki ˘ ı and M

Ju. Dalec ′ki ˘ ı and M. Kre ˘ ın,Stability of Solutions of Diﬀerential Equations in Banach Space , Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974

work page 1974

[11] [11]

Dragiˇ cevi´ c, A spectral characterization of nonuniform polynomial di- chotomies, Math

D. Dragiˇ cevi´ c, A spectral characterization of nonuniform polynomial di- chotomies, Math. Nachr., to appear

work page

[12] [12]

P. V. Hai, On the polynomial stability of evolution families , Appl. Anal. 95 (2016), 1239–1255

work page 2016

[13] [13]

Henry, Geometric Theory of Semilinear Parabolic Equations , Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981

D. Henry, Geometric Theory of Semilinear Parabolic Equations , Lecture Notes in Mathematics 840, Springer-Verlag, Berlin-New York, 1981

work page 1981

[14] [14]

Huy, Exponential dichotomy of evolution equations and admissib ility of func- tion spaces on a half-line , J

N. Huy, Exponential dichotomy of evolution equations and admissib ility of func- tion spaces on a half-line , J. Funct. Anal. 235 (2006), 330–354. ADMISSIBILITY AND POLYNOMIAL DICHOTOMIES 19

work page 2006

[15] [15]

Latushkin, T

Y. Latushkin, T. Randolph and R. Schnaubelt, Exponential dichotomy and mild solution of nonautonomous equations in Banach spaces , J. Dynam. Diﬀerential Equations 10 (1998), 489–510

work page 1998

[16] [16]

Li, Die Stabilit¨ atsfrage bei Diﬀerenzengleichungen, Acta Math

T. Li, Die Stabilit¨ atsfrage bei Diﬀerenzengleichungen, Acta Math. 63 (1934), 99–141

work page 1934

[17] [17]

Lupa and L

N. Lupa and L. Popescu, Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line , Semigroup Forum 98 (2019), 184– 208

work page 2019

[18] [18]

Massera and J

J. Massera and J. Sch¨ aﬀer,Linear diﬀerential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517–573

work page 1958

[19] [19]

Massera and J

J. Massera and J. Sch¨ aﬀer, Linear Diﬀerential Equations and Function Spaces , Pure and Applied Mathematics 21, Academic Press, New York-Londo n, 1966

work page 1966

[20] [20]

Megan, A

M. Megan, A. L. Sasu and B. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces , Integral Equations Operator Theory 44 (2002), 71–78

work page 2002

[21] [21]

J. S. Muldowney, Dichotomies and asymptotic behaviour for linear diﬀerenti al systems, Trans. Amer. Math. Soc. 283 (1984), 465–484

work page 1984

[22] [22]

Naulin and M

R. Naulin and M. Pinto, Roughness of (h, k)-dichotomies, J. Diﬀerential Equa- tions 118 (1995), 20–35

work page 1995

[23] [23]

Naulin and M

R. Naulin and M. Pinto, Stability of Discrete Dichotomies for Linear Diﬀerence Systems, J. Diﬀerence Equ. Appl. 3 (1997), 101–123

work page 1997

[24] [24]

Perron, Die Stabilit¨ atsfrage bei Diﬀerentialgleichungen, Math

O. Perron, Die Stabilit¨ atsfrage bei Diﬀerentialgleichungen, Math. Z. 32 (1930), 703–728

work page 1930

[25] [25]

Preda and M

P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc. 27 (1983), 31–52

work page 1983

[26] [26]

Preda, A

P. Preda, A. Pogan and C. Preda, ( Lp, Lq)-admissibility and exponential di- chotomy of evolutionary processes on the half-line , Integral Equations Operator Theory 49 (2004), 405–418

work page 2004

[27] [27]

Preda, A

P. Preda, A. Pogan and C. Preda, Sch¨ aﬀer spaces and exponential dichotomy for evolutionary processes , J. Diﬀerential Equations 230 (2006), 378–391

work page 2006

[28] [28]

A. L. Sasu, M. Babutia and B. Sasu, Admissibility and nonuniform exponential dichotomy on the half-line , Bull. Sci. Math. 137 (2013), 466–484

work page 2013

[29] [29]

A. L. Sasu and B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces , Integral Equations Operator Theory 54 (2006), 113–130

work page 2006

[30] [30]

A. L. Sasu and B. Sasu, Exponential trichotomy and p-admissibility for evolu- tion families on the real line , Math. Z. 253 (2006), 515–536

work page 2006

[31] [31]

A. L. Sasu and B. Sasu, Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications , Integral Equations Operator Theory 66 (2010), 113–140

work page 2010

[32] [32]

Van Minh, F

N. Van Minh, F. R¨ abiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equ ations on the half- line, Integral Equations Operator Theory 32 (1998), 332–353

work page 1998

[33] [33]

Zhou and W

L. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for diﬀerence equations , J. Funct. Anal. 271 (2016), 1087–1129

work page 2016

[34] [34]

L. Zhou, K. Lu and W. Zhang, Equivalences between nonuniform exponential dichotomy and admissibility , J. Diﬀerential Equations 262 (2017), 682–747. Department of Mathematics, University of Rijeka, Croatia E-mail address : ddragicevic@math.uniri.hr

work page 2017